First step is to realize that when you add up the three sides, the three vertices are being counted twice. So, the sum of the three vertices have to be (20×3) – sum(1…9) = 15.

That 15 can be split in many ways. For example, 4,5,6. Then the rest can be figured out (mostly by mental tries) as (2,9), (1,8), and (3,7).

Now, all combinations of vertices adding to 15 will not work.

I am not sure I have a way to tell which are the valid combinations.